When facing an ill-posed problem with unknown operator, few methods can be
applied. In particular, most of the regularization methods can not be used
since the knowledge of the eigenvalues of the operator is necessary. Hence
Bayesian method can provide a consistent method to build estimates for the
inverse problem. Here we focus on a Bayesian reconstruction of the inverse
function considered as a discrete measure. This method can be applied to
a large variety of fields and we will give as examples the cases of
seismic tomography and survey sampling.